Constraints on models for the initial collision geometry in ultra relativistic heavy ion collisions
Roy A. Lacey, Rui Wei, N. N. Ajitanand, J. M. Alexander, X. Gong, J. Jia, A. Taranenko, R. Pak, Horst Stocker
aa r X i v : . [ nu c l - e x ] M a y Constraints on models for the initial collision geometry in ultra relativistic heavy ion collisions Roy A. Lacey,
1, 2, ∗ Rui Wei, N. N. Ajitanand, J. M. Alexander, X. Gong, J. Jia,
1, 2
A. Taranenko, R. Pak, and Horst St¨ocker Department of Chemistry, Stony Brook University, Stony Brook, NY, 11794-3400, USA Physics Department, Bookhaven National Laboratory, Upton, New York 11973-5000, USA Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at D60438 Frankfurt am Main, Germany (Dated: October 30, 2018) Monte Carlo simulations are used to compute the centrality dependence of the collision zone ec-centricities ( ε , ), for both spherical and deformed ground state nuclei, for different model scenarios.Sizable model dependent differences are observed. They indicate that measurements of the 2 nd and4 th order Fourier flow coefficients v , , expressed as the ratio v ( v ) , can provide robust constraintsfor distinguishing between different theoretical models for the initial-state eccentricity. Such con-straints could remove one of the largest impediments to a more precise determination of the specificviscosity from precision v , measurements at the Relativistic Heavy Ion Collider (RHIC). PACS numbers: 25.75.-q, 25.75.Dw, 25.75.Ld Energetic collisions between heavy ions at the Rela-tivistic Heavy Ion Collider (RHIC), produce a stronglyinteracting quark gluon plasma (QGP). In non-centralcollisions, the hydrodynamic-like expansion of thisplasma [1–6] results in the anisotropic flow of particles inthe plane transverse to the beam direction [7, 8]. At mid-rapidity, the magnitude of this momentum anisotropy ischaracterized by the even order Fourier coefficients; v n = D e in ( φ p − Φ RP ) E , n = 2 , , .., (1)where φ p is the azimuthal angle of an emitted particle, Φ RP is the azimuth of the reaction plane and the brackets denote averaging over particles and events. The elliptic flow coefficient v is observed to dominate over the higher order coefficients in Au+Au collisions at RHIC ( i.e. v n ∝ ( v ) n and v <<
1) [9, 10]. The magnitudes and trends of v , are known to be sensitive to the transport properties of the expanding partonic matter [3, 4, 6, 11–17]. Consequently, there is considerable current interest in their use for quantitative extraction of the specific shear viscosity, i.e. the ratio of shear viscosity η to entropy density s of the plasma. Such extractions are currently being pursued via com- parisons to viscous relativistic hydrodynamic simulations [16–18], transport model calculations [14, 15] and hybrid approaches which involve the parametrization of scaling violations to ideal hydrodynamic behavior [10, 12, 13]. In all cases, accurate knowledge of the initial eccentricity ε , of the collision zone, is a crucial unresolved prereq- uisite for quantitative extraction of ηs . To date, no direct experimental measurements of ε , have been reported. Thus, the necessary theoretical esti- mates have been obtained by way of the overlap geometry corresponding to the impact parameter b of the collision, or the number of participants N part in the collision zone. A robust constraint for N part values can be obtained via measurements of the final hadron multiplicity or trans- verse energy. However, for a given value of N part , the theoretical models used to estimate ε give results which differ by as much as ∼
25% [19, 20] – a difference which leads to an approximate factor of two uncertainty in the extracted η/s value [16]. Therefore, an experimental con- straint which facilitates a clear choice between the differ- ent theoretical models is essential for further progress toward precise extraction of η/s . In ideal fluid dynamics, anisotropic flow is directly pro- portional to the initial eccentricity of the collision zone. A constant ratio for the flow coefficients v ( v ) ≈ . predicted [21]. It is well established that initial eccen- tricity fluctuations also influence the magnitude of v , significantly [5, 10, 21–23], i.e. the presence of these fluc- tuations serve to increase the value of v , . Therefore, one avenue to search for new experimental constraints, is to use ε , as a proxy for v , and study the model dependencies of their magnitudes and trends vs. N part . In this communication we present calculated results of ε , for collisions of near-spherical and deformed isotopes, for the Glauber [22, 24] and the factorized Kharzeev- Levin-Nardi (fKLN) [25, 26] models, i.e. the two primary models currently employed for eccentricity estimates. We find sizable differences, both in magnitude and trend, for the the ratios ε ( ε ) obtained from both models. This sug- gests that systematic comparisons of the measurements for the N part dependence of the ratio v ( v ) for these iso- FIG. 1. Calculated values of ε , vs. N part for MC-Glauber(open symbols) and MC-KLN (closed symbols) for Au+Aucollisions (a) and near-spherical Dy and deformed
Dyas indicated in (b). topic systems, can give direct experimental constraints for these models. Monte Carlo (MC) simulations were used to calculateevent averaged eccentricities (denoted here as ε , ) withinthe framework of the Glauber (MC-Glauber) and fKLN(MC-KLN) models, for near-spherical and deformed nu-clei which belong to an isobaric or isotopic series. Here,the essential point is that, for such series, a broad rangeof ground state deformations have been observed for rel-atively small changes in the the number of protons orneutrons [27, 28]. For each event, the spatial distribu-tion of nucleons in the colliding nuclei were generatedaccording to the deformed Woods-Saxon function: ρ ( r ) = ρ e (r − R (1+ β Y ( θ )+ β Y ( θ ))) /d , (2)where R and d are the radius and diffuseness parameters and β , are the deformation parameters which charac- terizes the density distribution of the nucleus about its polarization axis ( z ′ ). To generate collisions for a given centrality selection, the orientation of the polarization axis for each nucleus ( θ , φ and θ , φ respectively) was randomly chosen in the coordinate frame whose z axis is the beam direc- tion. For each collision, the values for N part and the number of binary collisions N coll were determined within the Glauber ansatz [24]. The associated ε , values were then evaluated from the two-dimensional profile of the density of sources in the transverse plane ρ s ( r ⊥ ), using modified versions of MC-Glauber [24] and MC-KLN [26] respectively. For each event, we compute an event shape vector S n and the azimuth of the the rotation angle Ψ ∗ n for n -th harmonic of the shape profile [29]; S nx ≡ S n cos ( n Ψ ∗ n ) = Z d r ⊥ ρ s ( r ⊥ ) ω ( r ⊥ ) cos( nφ ) , (3) S ny ≡ S n sin ( n Ψ ∗ n ) = Z d r ⊥ ρ s ( r ⊥ ) ω ( r ⊥ ) sin( nφ ) , (4) Ψ ∗ n = 1 n tan − (cid:18) S ny S nx (cid:19) , (5) where φ is the azimuthal angle of each source and the weight ω ( r ⊥ ) = r ⊥ . The eccentricities were calculated as: ε = h cos 2( φ − Ψ ∗ ) i ε = h cos 4( φ − Ψ ∗ ) i (6) where the brackets denote averaging over sources, as well as events belonging to a particular centrality or impact parameter range. For the MC-Glauber calculations, an additional entropy density weight was applied reflecting the combination of spatial coordinates of participating nucleons and binary collisions [19, 23] ; ρ s ( r ⊥ ) ∝ (cid:20) (1 − α )2 dN part d r ⊥ + α dN coll d r ⊥ (cid:21) , (7) where α = 0 .
14 was constrained by multiplicity measure- ments as a function of N part for Au+Au collisions [30]. The procedures outlined above (cf. Eqs. 2 - 7) ensure that, in addition to the fluctuations which stem from the orientation of the initial “almond-shaped” collision zone [relative to the impact parameter], the shape-induced fluctuations due to nuclear deformation are also taken into account. Note that ε , (cf. Eq. 6) correspond to v , measurements in the so-called participant plane [22, 24]. That is, the higher harmonic ε is evaluated relative to the principal axis determined by maximizing the quadrupole moment. This is analogous to the mea- surement of v with respect to the 2 nd order event-plane in actual experiments. One consequence is that the den- sity profile is suppressed, as well as the moment for the higher harmonic. Calculations were performed for a variety of isotopes and isobars with a broad range of known β , values. Here, we show and discuss only a representative set of results for Au ( R = 6 .
38 fm , β = − . , β = − . Dy ( R = 5 .
80 fm , β = 0 . , β = 0 .
00) and Dy ( R = 5 .
93 fm , β = 0 . , β = 0 .
06) [27, 28]. For these calculations we used the value d = 0 .
53 fm. Figure 1(a) shows a comparison of ε , vs. N part for MC-Glauber (open symbols) and MC-KLN (filled sym- bols) for Au+Au collisions. The filled symbols indicate larger ε , values for MC-KLN over most of the consid- ered N part range. The effect of shape deformation is il- lustrated in Fig. 1(b) where a comparison of ε , vs. N part [for MC-Glauber] is shown for the two Dy isotopes indicated. Both ε and ε show a sizable increase for the isotope with the largest ground state deformation FIG. 2. Comparison of ε ( ε ) vs. N part for near-spherical Dy (filled symbols) and deformed
Dy (open symbols)collisions. Results are shown for MC-Glauber (a) and MC-KLN (b) respectively. ( Dy). This reflects the important influence of shape- driven eccentricity fluctuations in collisions of deformed nuclei [31–34]. The magnitudes and trends of all of these eccentricities are expected to influence the measured val- ues of v , for these systems. A priori, the model-driven and shape-driven eccentric- ity differences shown in Fig. 1, need not be the same for ε and ε . Therefore, we present the ratio ε ( ε ) vs. N part , for both models in Fig. 2. The ratios obtained for Dy (near-spherical) and
Dy (deformed) with MC- Glauber are compared in Fig. 2(a); the same comparison is given in Fig. 2(b) but for MC-KLN calculations. Fig. ε ( ε ) for Dy and
Dy over the full range of N part considered. This difference stems from additional shape- driven fluctuations present in in collisions of Dy, but absent in collisions of Dy. The same comparison for MC-KLN results, shown in Fig. 2(b), points to a smaller difference for these ratios, as well as a different N part de- pendence. We attribute this to the difference in the trans- verse density distributions employed in MC-Glauber and MC-KLN. For a given value of N part , the measured ratio of the flow coefficients v ( v ) for Dy+
Dy and
Dy+ Dy collisions, are expected to reflect the magnitude and trend of the ratio ε ( ε ) (note that a constant ratio ≈ . is predicted for ideal hydrodynamics without the influ- ence of fluctuations [21]). Fig. 2 suggests that a rela- tively clear distinction between fKLN-like and Glauber- like initial collision geometries could be made via system- atic studies of v ( v ) for near-spherical and deformed iso- topes/isobars. Specifically, a relatively smaller (larger) FIG. 3. N part dependence of ε ( ε ) (a), ε ( m )( ε ( m )) (b) and R( m )(R) (c) for Au+Au collisions (see text). The open and closedsymbols indicate the results from MC-Glauber and MC-KLNrespectively. difference between the ratios v ( v ) for each isotope, would be expected for fKLN (Glauber) initial geometries. Sim- ilarly the scaling of v , data from the isotopic or isobaric pair would be expected only for MC-Glauber or MC-KLN eccentricities. Note that the influence of a finite viscosity is expected to be the same for both systems and therefore would not change these conclusions. The filled symbols in Figs. 2 (a) and (b) also suggest a substantial difference in the ε ( ε ) ratios predicted by MC-Glauber and MC-KLN respectively, for collisions be- tween near-spherical nuclei. This difference is also appar- ent in Fig. 3(a) where the calculated ratios for Au+Au ( β = − . , β = − .
03) collisions are shown. The MC-KLN results (filled circles) indicate a relatively flat dependence for 40 . N part . the characteristic decrease, for the same N part range, seen in the MC-Glauber results. As discussed earlier, each of these trends is expected to influence the measured ratios of the flow coefficients v ( v ) . Therefore, an experimental observation of a rel- atively flat N part dependence for v ( v ) [over the range . N part . collision geometries in Au+Au collisions. Such a trend has been observed in the preliminary and final data sets reported in Refs. [10, 21, 35] and is consistent with the conclusions reached in Ref. [10, 36] that the N part and impact parameter dependence of the eccentricity scaled flow coefficients v ε and v ε favor fKLN-like initial collision geometries. The closed symbols in Figs. 2(b) and 3(a) indicate a decreasing trend for ε ( ε ) for near-spherical nuclei for N part & fact that, in each event, ε is computed in the reference frame which maximizes the quadrupole shape distribu- tion, i.e. the so-called participant frame. In this frame, ε can take on positive or negative event-by-event val- ues. Consequently, smaller mean values are obtained, especially in the most central collisions. Fig. 2 shows that the relatively large ground state deformation for Dy (open symbols) leads to an increase of ε ( ε ) [rela- tive to that for the spherical Dy isotope] which is espe- cially pronounced in the most central collisions. However, Fig. 3(a) shows that the modest deformation for the Au nuclei does not lead to a similarly increasing trend for N part &
200 as implied by data [21, 35]. The relatively flat N part dependence for v ( v ) , over the range 40 . N part .
200 in Fig. 3(a), suggests fKLN-like collision geometries. Consequently, it is interesting to in- vestigate whether or not the magnitude of the ratios for N part & pact on the values for N part . that a large increase of ε ( ε ) can indeed be obtained for N part &
200 with relatively little change in the magni- tude and trend of the ratios for N part . achieved by introducing a correlation or mixing ( m ) be- tween the principal axes of the quadrupole (Ψ ∗ ) and hex- adecapole (Ψ ∗ ) density profiles associated with ε and ε respectively. That is, the orientation of Ψ ∗ was modified to obtain the new value Ψ ∗∗ = (1 − γ )Ψ ∗ + γ Ψ ∗ , where γ = 0 .
2. This procedure is motivated by the finding that, in addition to the v contributions which stem from the initial hexadecapole density profile, experimental mea- surements could also have a contribution from v [with magnitude ∝ ( v ) ] [29, 37]. The correlation has little, if any, influence on the ε values, but does have a strong influence on ε ( ε ) in the most central collisions. This is demonstrated in Fig. 3(c) where the double ratio R( m )R (R( m ) = ε ( m )( ε ( m )) and R = ε ( ε ) ) is shown. In summary, we have presented results for the initial eccentricities ε , for collisions of near-spherical and de- formed nuclei, for the two primary models currently em- ployed for eccentricity estimates at RHIC. The calculated ratios for ε ( ε ) , which are expected to influence the mea- sured values of v ( v ) , indicate sizable model dependent differences [both in magnitude and trend] which can be exploited to differentiate between the models. The ε ( ε ) ratios obtained as a function of N part for Au+Au colli- sions with the fKLN model ansatz, show trends which are strongly suggestive of the measured ratios for v ( v ) observed in Au+Au collisions for 40 . N part . For more central collisions ( N part & trend is strongly influenced by initial eccentricity fluctu- ations if a correlation between the principal axes of the quadrupole and hexadecapole density profiles is assumed. New measurements of v ( v ) for collisions of near-spherical and deformed isotopes (or isobars) are required to exploit these tests. Acknowledgments
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